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Uniform prox-regular sets in Hilbert setting Uniform prox-regularity of constraint sets Application to constrained optimization Inverse image, intersection Preservation of uniform prox-regularity of sets and application to constrained optimization Florent Nacry Joint work with Samir Adly and Lionel Thibault, accepted for publication in SIOPT. GDR MOA Université de Bourgogne 2015 December 2 Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Notations, definitions Uniform prox-regularity of constraint sets Chebyshev set Application to constrained optimization Clarke tangent cone, normal cone and subdifferential Inverse image, intersection Uniform prox-regularity Plan Uniform prox-regular sets in Hilbert setting 1 Notations, definitions Chebyshev set Clarke tangent cone, normal cone and subdifferential Uniform prox-regularity Uniform prox-regularity of constraint sets 2 An introduction to the problem Semiconvexity Nonsmooth Inequalities Smooth equalities Smooth inequalities and equalities Application to constrained optimization 3 Inverse image, intersection 4 Intersection Inverse image Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Notations, definitions Uniform prox-regularity of constraint sets Chebyshev set Application to constrained optimization Clarke tangent cone, normal cone and subdifferential Inverse image, intersection Uniform prox-regularity Notations, definitions � Let H be a real Hilbert space, �· , ·� be the inner product and �·� = �· , ·� be the associated norm. . Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Notations, definitions Uniform prox-regularity of constraint sets Chebyshev set Application to constrained optimization Clarke tangent cone, normal cone and subdifferential Inverse image, intersection Uniform prox-regularity Notations, definitions � Let H be a real Hilbert space, �· , ·� be the inner product and �·� = �· , ·� be the associated norm. • B H = { x ∈ H : � x � ≤ 1 } . Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Notations, definitions Uniform prox-regularity of constraint sets Chebyshev set Application to constrained optimization Clarke tangent cone, normal cone and subdifferential Inverse image, intersection Uniform prox-regularity Notations, definitions � Let H be a real Hilbert space, �· , ·� be the inner product and �·� = �· , ·� be the associated norm. • B H = { x ∈ H : � x � ≤ 1 } . • For S ⊂ H , co S is the convex hull of S , bdry S is the boundary of S . Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Notations, definitions Uniform prox-regularity of constraint sets Chebyshev set Application to constrained optimization Clarke tangent cone, normal cone and subdifferential Inverse image, intersection Uniform prox-regularity Notations, definitions � Let H be a real Hilbert space, �· , ·� be the inner product and �·� = �· , ·� be the associated norm. • B H = { x ∈ H : � x � ≤ 1 } . • For S ⊂ H , co S is the convex hull of S , bdry S is the boundary of S . • For / 0 � S ⊂ H , for all x ∈ H s ∈ S � x − s � Proj S ( x ) = { y ∈ S : d S ( x ) = � x − y �} . d S ( x ) = inf and For each x ∈ H , when Proj S ( x ) contains one and only one vector y ∈ H , we set P S ( x ) = y . Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Notations, definitions Uniform prox-regularity of constraint sets Chebyshev set Application to constrained optimization Clarke tangent cone, normal cone and subdifferential Inverse image, intersection Uniform prox-regularity Chebyshev set Let H be a real Hilbert space, S a nonempty (strongly) closed subset of H . Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Notations, definitions Uniform prox-regularity of constraint sets Chebyshev set Application to constrained optimization Clarke tangent cone, normal cone and subdifferential Inverse image, intersection Uniform prox-regularity Chebyshev set Let H be a real Hilbert space, S a nonempty (strongly) closed subset of H . Problem: • Assume that for all x ∈ H , Proj S ( x ) is a singleton (i.e., S is a Chebyshev set). Is the set S convex ? Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Notations, definitions Uniform prox-regularity of constraint sets Chebyshev set Application to constrained optimization Clarke tangent cone, normal cone and subdifferential Inverse image, intersection Uniform prox-regularity Chebyshev set Let H be a real Hilbert space, S a nonempty (strongly) closed subset of H . Problem: • Assume that for all x ∈ H , Proj S ( x ) is a singleton (i.e., S is a Chebyshev set). Is the set S convex ? Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Notations, definitions Uniform prox-regularity of constraint sets Chebyshev set Application to constrained optimization Clarke tangent cone, normal cone and subdifferential Inverse image, intersection Uniform prox-regularity Chebyshev set Let H be a real Hilbert space, S a nonempty (strongly) closed subset of H . Problem: • Assume that for all x ∈ H , Proj S ( x ) is a singleton (i.e., S is a Chebyshev set). Is the set S convex ? • T. Motzkin, L.N.H. Bunt, M. Kritikos (1930-1940): Dim H < + ∞ ⇒ S is convex. Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Notations, definitions Uniform prox-regularity of constraint sets Chebyshev set Application to constrained optimization Clarke tangent cone, normal cone and subdifferential Inverse image, intersection Uniform prox-regularity Chebyshev set Let H be a real Hilbert space, S a nonempty (strongly) closed subset of H . Problem: • Assume that for all x ∈ H , Proj S ( x ) is a singleton (i.e., S is a Chebyshev set). Is the set S convex ? • T. Motzkin, L.N.H. Bunt, M. Kritikos (1930-1940): Dim H < + ∞ ⇒ S is convex. • Dim H = + ∞ ⇒ Still open. With additional assumptions (L.P . Vlasov, E. Asplund, V. Klee, ...) ⇒ True Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Notations, definitions Uniform prox-regularity of constraint sets Chebyshev set Application to constrained optimization Clarke tangent cone, normal cone and subdifferential Inverse image, intersection Uniform prox-regularity Open enlargement Let H be a real Hilbert space and r ∈ ]0 , + ∞ ] an extended real. For S a nonempty closed subset of H , one defines U r ( S ) = { x ∈ H : d S ( x ) < r } , the r-open enlargement of S . Problem: • What is the class of nonempty closed subsets S of H such that Proj S ( x ) is a singleton for all x ∈ U r ( S ) and the well-defined mapping P S : U r ( S ) −→ H is continuous ? Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Notations, definitions Uniform prox-regularity of constraint sets Chebyshev set Application to constrained optimization Clarke tangent cone, normal cone and subdifferential Inverse image, intersection Uniform prox-regularity Clarke tangent cone, normal cone and subdifferential Definition Let H be a real Hilbert space. The Clarke tangent cone (resp. Clarke normal cone ) to a subset S ⊂ H at x ∈ S is the set T C ( S ; x ) = { h ∈ X : ∀ S ∋ x n → x , ∀ t n ↓ 0 , ∃ X ∋ h n → h , x n + t n h n ∈ S , ∀ n ∈ N } (resp. N C ( S ; x ) = � v ∈ H , � v , h � ≤ 0 , ∀ h ∈ T C ( S ; x ) � ) . For x ∈ H , U an open neighborhood of x and f : U −→ R an extended real-valued function finite at x , the Clarke subdifferential of f at x is defined as the set � v ∈ H : ( v , − 1) ∈ N C ( epi f ;( x , f ( x ))) � ∂ C f ( x ) = . Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity

Uniform prox-regular sets in Hilbert setting Notations, definitions Uniform prox-regularity of constraint sets Chebyshev set Application to constrained optimization Clarke tangent cone, normal cone and subdifferential Inverse image, intersection Uniform prox-regularity Clarke subdifferential in finite dimension Theorem Let m ∈ N , f : R m −→ R be Lipschitz continuous near x ∈ R m , △ f the subset of R m where ∇ f exists, D f any subset of △ f such that △ f \ D f is Lebesgue negligible. Then, one has � � ∂ C f ( x ) = co n → + ∞ ∇ f ( x n ) : D f ∋ x n → x lim . Florent Nacry, joint work with Samir Adly and Lionel Thibault Preservation of uniform prox-regularity